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JOURNAL 



OF THE 



INSTITUTE OF ACTUARIES 



AND 



ASSUKANCE MAGAZINE. 



Some Account of James Dodson, F.R.S. By A. Be Morgan, Esq. 

JN life has been written of the original projector of the Equitable 
Society, except in a column of the Biographie Universelle by 
M. Nicollet. Dodson's name was, and even still is, so familiar 
to the actuary, chiefly through the Mathematical Repository, and 
the impulse he gave to life-contingency problems, that this Journal 
is the proper place of deposit for what can be collected concerning 
him. The article above mentioned tells very little. He succeeded 
Hodgson [which should have been Robertson] in the chair of 
mathematics at Christchurch Hospital in 1756 [1755] and died 
November 23, 1757. He published the Antilogarithmic Canon, 
which others had contemplated [and executed too, but the manu- 
script was lost] and which he had the courage to execute up to a 
certain point [his table is the counterpart of Vlacq's largest direct 
table : five figures of argument and eleven of tabular result]. He 
could not balance the success of the ordinary tables : the writer 
doubts whether the table was ever used on the continent [he might 
have added, England : who uses either Vlacq or Dodson ? Their 
tables are for help to other table-makers, and always were, though 
both of them intended more]. He published the Calculator in 
1747, a collection of tables at the end of which [say in the proper 
VOL. xiv. 2 B 



342 Some Account of James Dodson, F.R.S. [Oct. 

place in the middle] is an abridgment of the antilogarithmic table. 
But he is best known in England by his Mathematical Repository, 
and by his zeal for benevolent institutions [say his determination 
to found an assurance office to which himself should be admissible] . 
In his lectures at the school of Christ-church Hospital he gave the 
first idea of a company for life assurance, a plan afterwards executed 
by Edw. Kowe Mores [and others] under the name of the Equit- 
able Society [I may safely contradict the statement that he lectured 
on life-assurance to the young men whom he was to instruct in 
mathematics and navigation]. 

James Dodson was my mother's father's father. All know- 
ledge of him was completely cut off from his posterity by his 
leaving no near relation, no widow, and no child above fourteen 
years of age. I have, in several cases, found biographical inquiry 
arrested by similar circumstances. He seems to have had but two 
children, both sons. One, the elder, reared a large family, and 
must have, by this time, upwards of & hundred and fifty descen- 
dants, dead and alive : but never more than one male descendant 
of the name in my generation. So much for the efficacy of a 
large preponderance of daughters in preserving a surname. Of 
the other, nothing was ever remembered except that he " gave his 
brother much trouble." With all my inquiry, curious as I was to 
know all about this ancestor, I never obtained from his family 
more than three pieces of information : the date of his eldest son's 
birth; a copy of his treatise on book-keeping, which seems to 
have been preserved by his son, and which was given to me by one 
of his grandsons; a tradition that he was befriended on some 
occasion by the Duke of Manchester, who I have no doubt was a 
misnomer of the Earl of Macclesfield. 

This paper is a case of the problem of constructing an unknown 
biography out of materials equally common to all mankind : and a 
sketch of a career may be given, as complete as many which are 
taken from contemporary record, and of much better evidence as 
to the separate facts than the unsupported statement of a casual 
writer. What may be done by one who takes the interest of a 
descendant in the matter is equally possible to be done by others : 
and a person who systematically collects all the biographical facts 
he meets with may find himself in a condition to give no small 
number of accounts, sufficient for literary purposes, of persons 
whose lives have been neglected. 

James Dodson must have been born shortly before 1710 : who 
he was, or from whence, I never found the slightest information. 



1868.] Some Account of James Dodson, F.R.S. 343 

From their long and close intimacy it must be suspected that he 
was a contemporary, perhaps a schoolfellow, of John Robertson 
(born 1712) the author of the Navigation, who in the history of 
Christ's Hospital is called the brother of Robertson the historian. 
But the following is more direct. In 1756, he found he could not 
assure in the Amicable Society, being over age : their limit was 45 ; 
and all accounts imply that he had but just passed the limit. He 
must have had some sort of liberal education ; for his use of the Ber- 
noullis, Euler, Ozanam, &c, shows that he read Latin and French. 
He must have been thrown on the world with some little command 
of money. He was able to spend unprofitable years in the con- 
struction of his antilogarithmic table, which he published on his 
own account in 1742 : it was his first public appearance. A pub- 
lisher's name (Wilcox) is joined with his own in the imprint : but 
we may be pretty sure that a folio of new tables at £1. 2s. 6d. 
(afterwards reduced to 12s.) by a young man quite unknown, would 
not find a publisher to take any risk. Those whom he mentions 
as his friends are Robertson, William Jones, of whom presently, 
and Labelye, who was, I believe, then building Westminster 
Bridge. Again, he had been, as we shall see, a pupil of De Moivre, 
who was at the top of the tree, and who must have been, at the 
time of Dodson' s pupillage, very well remunerated, as one of the 
most famous of mathematicians, and Newton's particular friend. 
Between the Canon and the next work on his own account, he 
added a wife to his means of expenditure, which looks as if the 
money were not quite gone. He must have married soon after the 
publication of the Canon, for his first son was born in 1743. He 
was, I suppose, an amateur worker up to this time : for he is not 
called 'teacher of the mathematics' in the title of the Canon, 
though, had he been thus employed, the advertisement would have 
been a very good one : it first appears in 1747. 

I have said he was a pupil of De Moivre. This is attested by 
Matthew Maty (M.D., afterwards Sec. R. S.) in his life, which is 
very little known, of his most particular friend De Moivre. Maty 
gives, as specimens of the pupils, Macclesfield, Cavendish, Stan- 
hope, Martin Folkes, Fatio de Duillier, Scot, Daval, and Dodson. 
Of Lord Macclesfield I need say nothing ; nor of Stanhope (the 
well known Lord Chesterfield), Folkes and Duillier. Cavendish 
was probably Lord Charles Cavendish, the father of the great 
chemist. Scot[t] was probably one of two fellows of the Royal 
Society of that name. Daval was a noted lawyer of a scien- 
tific turn, no doubt the Peter Daval who became Secretary 

2 b 2 



344 Some Account of James Dodson, F.R.S. [Oct. 

of the Royal Society in 1759. Those who look into the history of 
the time will find evidence of a Be Moivre clique, kept together 
by intercourse with their old teacher, who lived until 1754, and 
by Maty's interest in the pupils of his old friend. Maty edited 
the Journal Britannique, a London publication in French, which 
expired shortly after De Moivre's death, living long enough to 
contain the biography mentioned, which was soon published sepa- 
rately. When any one of the pupils published a work, it was 
immediately favourably reviewed. When Sam Johnson's dic- 
tionary appeared, the review suppressed all about the celebrated 
letter to Lord Chesterfield, and hinted that Johnson should not 
have cast off the patron he himself had chosen at the beginning. 
So Johnson said of Maty, " He ! the little black dog ! I would 
throw him into the Thames :" from this we draw an inference which, 
in some very grave and dignified dictionary, will one day appear 
as " We have the testimony of the celebrated Dr. Johnson that 
Maty was short and dark : some take the great lexicographer as 
saying that he was of a surly temper, and not so much given to 
ablution as would in our time be held desirable ; but we doubt if 
we can safely adopt this interpretation." 

Various relations between the pupils are found. Lords Mac- 
clesfield and Chesterfield moved and seconded the second reading 
of the change of style ; and Daval drew the bill. Dodson dedi- 
cated to his old teacher, and to the two peers ; by whom he was 
also employed in surveying and accounts. I trace him through 
his writings as a private teacher, accountant, surveyor, &c, pro- 
bably an answerer of actuary's cases, until 1755, when he gained 
what was for him a splendid rise in the world. 

Charles II., who was a dabbler in science, and sometimes in a 
more creditable way than assisting at the joke of dissecting the 
body of an infant picked up about the palace, — and who really had 
that sense of the importance of navigation which an English 
Sovereign ought to have, — founded three Royal Institutions : the 
Royal Society, the Royal Observatory, and (1673) the Royal 
Mathematical School, attached to Christ's Hospital, for mathematics 
and navigation. The " New System of Mathematics" (2 vols. 
4to. 1681) was written for this school by Jonas Moore, Master 
of the Ordnance, by whose advice it was founded : the course was 
left not quite finished, and Halley and Flamsteed took part in its 
completion. This school has always been distinct from the ordi- 
nary teaching of the Hospital, being especially devoted to navi- 
gation: and I have seen an elementary work announced as 



1868.] Some Account of James Dodson, F.R.S. 345 

intended for both the schools. At first the teachership was an 
office of very high consideration. "When the Royal Society nomi- 
nated Halley on the committee for keeping Newton to his work 
(the Principia) or as they phrased it, to " keep Mr. Newton in 
mind of his promise," the second member, who had a mere sine- 
cure, would certainly be a person whose position was a guarantee 
for most respectful meaning on the part of the Society ; especially 
considering the curious nature of the duty. Except in this instance 
I never heard of a scientific body extorting a promise that a book 
should be written, and appointing a committee to see that it was 
done. This second member was Mr. Paget, or Pagett, master 
of the Royal Mathematical School : and that he was selected for 
his position rather than his merits I infer from his carelessness as 
a teacher being notorious; he afterwards took to drinking, or 
perhaps we should say that his having taken to drinking after- 
wards became as notorious as his neglect of his duties. The post 
gradually declined in external notoriety, as the Royal School — 
which still exists — was more and more nearly absorbed into the 
Hospital. Very few of those who hear of the boys annually 
presenting their charts, &c, for the inspection of the Sovereign 
are aware that this privilege belongs to the Royal Mathematical 
School, and not to the Blue Coat School itself. It may be gathered 
from various circumstances that the post was, in 1755, no mean 
addition of station to the private teacher who had lived by all 
kind of odd jobs at " the Blue Legg, near to Bell Dock, Wapping." 
He gained it, as I suppose, by the influence of Lord Macclesfield, 
who was then President of the Royal Society : I thus interpret the 
imperfectly remembered tradition of a granddaughter, that he was 
befriended by the Duke of Manchester. He was admitted of the 
Royal Society Jan. 23, 1755, which was probably before his 
appointment to the teachership in the same year. This is fully 
confirmed by the third volume of the Repository. The preface is 
dated Jan. 23, 1755, which means that he had waited to date his 
preface until he could put F.R.S. after his name: a precedent 
for the Society, should it ever want one, that the admission, not 
the election (which had taken place a week before) gives the 
literary character. But he is not styled as of the R.M.S. : only 
" accomptant and teacher of the mathematics." The little point 
is to the following purpose. The Royal Society was somewhat 
exclusive during the last century, and rather averse to admit men 
in trade. But we must infer that Dodson was not elected because 
his new post made him grand enough, but that he might become 



346 Some Account of James Dodson, F.R.S. [Oct. 

grand enough for the (prohably) promised post. His friend 
Robertson, who preceded him, and whose position exactly resem- 
bled his own, had been F.R.S. since 1741 : he held the post only 
about a year. And Hodgson, who came before Robertson, had 
been in the Society since 1703, five years before he gained the 
mastership. Accordingly, it seems to have been the rule to fill up 
the place from among the fellows of the Royal Society : but several 
of Dodson's early predecessor^ came into the school first, and into 
the Society shortly afterwards. 

Dodson, thus comparatively enriched and established, wanted 
to insure his life, and found that the Amicable received no lives 
over 45. He accordingly set himself (1756) to found a new office ; 
and thus became the projector of the Equitable Society, as pre- 
sently described. Thomas Simpson was lecturing on the subject, 
with a view to a new office : Dodson called a meeting by advertise- 
ment, and formed a Committee. I find no trace of concert. I 
suspect that Simpson was looked on coldly by the De Moivre clique : 
many know the savage onslaught made by De Moivre on Simpson, 
though it seems the assailant afterwards cooled down. But it may 
be suspected that respect for the old man who represented the 
school of Newton, Leibnitz, the Bernoullis, &c, so long after they 
were gone, prevented much mention of Simpson, whom I do not 
find prominently cited by Dodson until after De Moivre's death, 
when he is spoken of in proper terms. A manuscript lecture of 
the period was lent to me many years ago, which showed no sign 
of being either by Simpson or by Dodson. Perhaps the plan was 
stirred in several quarters. 

Dodson must have found his position very troublesome. His 
pupils were about twenty years of age: and the mixture of these 
men with the boys of the school led to all kinds of disturbance, 
beginning probably with interchange of chaffing and cuffing. But 
he did not enjoy it long; he died November 23, 1757. He leaves 
the character of a useful mathematician, inventive in application, 
but not in augmentation, of his science. He was eminently 
effective, and this until long after his death, — indeed, until 1820 
at least — in attracting the attention of students of annuities and 
assurance to the problems connected with their subject. His term 
of public life was only fifteen years : and he was of a period in 
which the study of pure mathematics in England was at the lowest 
ebb. Had a man of thirty-two years old emerged from obscurity 
in the early time of Newton with such a folio as the Canon, no 
doubt the work of years, he would have been noticed and 



1868.] Some Account of James Dodson, F.R.S. 347 

encouraged : but nothing of this sort took place. To get an idea 
of our state at the time, say 1740-1760, take the names of all who 
were alive in Britain, no matter at what age, in any part of that 
period, and who can in any way be identified with pure mathema- 
tics. We have the remains of the old school, Berkeley, De Moivre, 
Halley, Jurin, Maclaurin, Robins, Stirling, William Jones and 
Braickenridge ; a powerful list. To them we may add Thomas 
Simpson, Matthew Stewart, Walmesley, Waring, Robert Simson, 
Atwood, Hutton, Emerson, Horsley, Maseres, Playfair, Judge 
Wilson. Dodson, then, is one of the larger stars of his constellation : 
but the constellation not one of first-rate brilliancy. Reuben 
Burrow would have been added to my list, if he had published 
anything of sufficient note: but he appears in another way. 
Again, look forward to 1807,' when we should see the crop of the 
seed-time just examined. In Mr. Walker's group — published six 
years ago — of fifty-one men of science of that day, the only two 
who are at all associated with pure mathematics are Leslie and 
Playfair. 

An inquiry into the state of mathematical studies at Cambridge 
would probably confirm what I have said. Before such men as 
Waring, Paley, Milner, Vince, &c, gave strength to the system, I 
suspect that it was much debilitated. Taking the general results 
of senior wranglership as one test, there is little to speak of until 
the effect of those I have named began to be seen : and then we 
have such phenomena as three years which produced two bishops 
and a lawyer of celebrity, followed by five years which produced 
four judges. Of the dead period I have but one anecdote which I 
know to be true : it will look much like caricature. The senior 
wrangler of 176- was in 1825 still resident in his college, and of 
course very old. He recommended a young candidate for honours, 
in presence of one from whom I heard it, to be sure to attend 
particularly to quadratic equations : it was a quadratic, said he, 
which made me senior wrangler. 

Any degree of celebrity, small or great, is not fairly established 
until detraction is proved : but this confirmation, as to Dodson, 
only turned up in our own day. The private diary of Reuben 
Burrow, a good mathematician, but eminently scurrilous and 
slanderous, is the place of deposit. For ample proof of this 
character, see the English Cyclopaedia ' Tables/ and also Notes and 
Queries, Series I. vol. xii. p. 142 and Series III. vol. v. pp. 107, 
215, 261, 303, 361. Burrow did not come into rivalry with 
Dodson, who is therefore let off cheaply : but poor Wales, against 



348 Some Account of James Dodson, F.R.S. [Oct. 

whom Burrow was an unsuccessful competitor on more occasions 
than one, particularly for Dodson's old place in 1775, is, with 
another, "two of the most stupid and most dirty of all possible 
fools, rogues, and scoundrels," while Wales, he by himself he, is 
"not only the dirtiest scoundrel that God ever made, but the 
dirtiest rascal that he possibly could make. Amen." This is in 
the fly leaf of a book in my possession : my reader will not need 
me to tell him that Wales was an irreproachable man. From the 
diary it appears that this character is not entirely given on scientific 
grounds : for the wife of the said Wales is chai'ged with having 
been the person who circulated the story that the said Burrow had 
given his own wife black eyes, a likely thing per se. The diary 
states that Wm. Jones, the father of the Indian Judge, so 
celebrated for his library and for his allowance of its use — the 
liberty of his study, Dodson calls it — as well as for his wide 
acquaintance with the mathematicians, was exceedingly rough and 
uncourteous :■•" Gardiner, the logarithm fellow, and Dodson, he 
used to treat like a couple of dogs." This is against all evidence 
of Jones's character : and I mention it first to note that Burrow 
calls Jones the Secretary of the Royal Society, which he never was ; 
and gives Robertson — who was then clerk of the Society — as his 
informant ; who must have known better, and who may safely be 
set down as never having said so. Probably Burrow confounded 
his man with Jezreel Jones, who was clerk of the Society, 
1698-1713. William Jones was a Welchman, brought up in 
Wales : and a certain irascibility is held to belong to the 
national character. In that day, it must be remembered, the 
temperament of the races was much more pronounced than in 
our day, in which it would be easy to pick out and bring together 
an Englishman, a Welchman, a Scotchman, and an Irishman, of 
whom a fifth person, after hearing them talk for an hour, would be 
puzzled ±o say which was which. It may be held credible that 
Jones occasionally flew out : and his genial disposition, which led 
him to lay his treasures open to all, especially to the young 
aspirants whom he was so ready to advise and assist, probably had 
two warm sides, one at each end. But he had passed a life among 
his superiors both in station and in science, and all the probabilities 
of the case, as well as general evidence, are against his having had 
any reputation for habitual roughness. No name of the period has 
come down to us in a clearer atmosphere of respect and esteem. 
Burrow then gives the following account (Aug. 22, 1775). 



1868.] Some Account of James Dodson, F.R.S. 349 

" I had a good deal of talk with Mr. Robertson, and staid supper. 
Ho told me that Mr. Wm. Jones wrote that history of logarithms prefixed 
to Dodson's tables of the Anti-logarithmic Canon : that Dodson wrote such 
a confused and odd style that there was neither head nor tail in it, hardly ; 
and that he himself drew up the examples. He also gave me the history 
of the Mathematical Repository, as follows. Mr. Robertson having taught 
General Conway mathematics (who was then only a colonel), after he was 
member of parliament he called on Mr. IS. and told him that as his place in 
the House hindered his further attendance to mathematical subjects he 
should drop it, but at the same time he should be glad to have those papers 
which he had learnt copied over. Mr. R. not having time or inclination to 
do this himself applied to Dodson. Dodson employed one Ralph to copy 
them, but at the same time Dodson took a copy for himself (which by the 
bye was a dirty action). This Mr. R. did not know to a long time after, 
when, happeniug to think on the scheme of publishing a mathematical 
repository, the first volume of which was to contain a volume of algebraical 
questions, and the second geometrical, he proposed it to Dodson, who 
readily accepted the offer of joining with him. This Mr. Robertson 
mentioned to Mr. Jones, but Mr. Jones told him he was against the affair 
on account of Mr. R.'s probability of publishing some of the methods Jones 
had taught him, which he (Jones) might have thought of publishing 
afterwards himself. Mr. R. on this set the affair aside himself, but 
Dodson went on with it, and the greatest part of the questions in the first 
volume, at least 200 of the questions, were copied from Mr. Robertson's 
papers. 

It will be worth while to follow up Mr. Burrow, because diary 
stories have been much relied on, and it will be instructive to point 
out what their value may be. I will therefore take one of a 
different kind, also derived from Robertson, upon which, as it 
happens, we are probably able to confront Burrow with Robertson 
himself. N.B. The blanks are not Burrow's. 

He [Lord Macclesfield] married a , his family were in confusion, 

and when he died the ordered all his papers to be burnt but such as 

related to money matters, and Jones (sic) papers never was (sic) seen nor 
heard of more. Some think they were burnt among the rest, but Horsfall, 
of the Temple, who was one of those employed, says there were no such 
papers among those that were burnt. Others say that a number of papers 
were sent down to Shirborne Castle in his lifetime 

The two octavo volumes of Macclesfield Correspondence — 
which are but a small portion of the manuscripts now at Shirburn 
Castle — refute the tale of the burning. And now as to the 
character of the second Lady Macclesfield. Lord M. is described 
as having married, in 1757, 'Dorothy, daughter of — Nesbitt.' 
This short description probably indicates that her family was not 
of rank or note : but I can find nothing against her in the scandal 
of the time. Burrow must have mixed her up with another story, 



350 Some Account of James Dodson, F.R.S. [Oct. 

which he probably did hear from Robertson, who left a slight 
written account of William Jones among his papers, which Hutton 
published. Robertson seems to have been one of those retailers of 
half-told stories who leave their hearers to fill up in their own way. 
Jones was the director of Lord Macclesfield's education until the 
young man travelled in France and Italy: and Robertson says 
"They tell a story of an Italian wedding, which caused great 
disturbance in Lord Macclesfield's family, but was compromised by 
Mr. Jones ; which gave rise to a saying that Macclesfield was the 
making of Jones and Jones the making of Macclesfield." The 
compromise of a wedding was a thing which might have happened 
in those days, when the marriage-law* of England was the old law 
of Europe, which we now call the Scotch law. If the story have 
any foundation the young lord must have made some exchange of 
declarations in Italy, with a woman who followed him to England, 
and Jones may have been employed in buying her off. This seems 
somewhat supported by the haste with which a wife was found for 
the young man, who set out on his travels about 1720, and was 
married to his first wife in 1722. Probably Burrow has spoilt the 
point of the epigram by reversing the points : if Jones extricated 
the son, and the father afterwards gave him a good place, it would 
have been that Jones was the making of Macclesfield, and (then) 
Macclesfield was the making of Jones. But probably the reference 
is to some place given by the son, in addition jto those already 
given by the father: Jones was certainly " made" long before 1720. 
I now go on to what directly concerns Dodson, who says he 
got his questions out of mathematicians of the two centuries 
preceding, of whom he names twenty-one. Burrow saye that 
more than 200 were exercises given by Robertson to General 
Conway, whom no one will believe to have mastered any 200 that 
can be pointed out. It is not credible that Dodson, himself a 
teacher, and a large importer of new algebra into a new subject, 
should have found it necessary to crib the simple equations, &c, 
of another teacher. Nor do I believe that Robertson told any 
such story of his friend past, present and future; especially to 
such a person as he knew Burrow to be. No doubt he told 
Burrow something: and Burrow had a power of inference not 

* There was in England an inveterate popular belief, without any foundation in law, 
that the declarations which made a marriage must be made before a person in orders, 
English or Roman. There is a great deal of confusion on this subject, in great part 
arising from not remembering that the marriage by declaration before witnesses, which 
was binding both civilly and ecclesiastically, was held irregular by the Church, and made 
the parties subject to spiritual censures and penances; and also to some statutory penalties, 
which were seldom or never enforced. 



1868.] Some Account of James Dodson, F.R.S. 351 

given to all. It was one of those eccentricities of genius in private 
life — to use the phrase of a biographer — by which he was as much 
distinguished as by his nihil quod tetigit non d — navit. Lord Howe 
did not convoy the India fleet until they were out of (Burrow's) 
fear of the French : so it is laid down that " he and his brother 
are a couple of cowardly scoundrels, or else that they are bribed by 
the enemy." This was followed by what was perhaps the nearest 
approach Burrow's mind could make to Domine, salvam fac patriam, 

but worded thus — "What d d stupidity this cursed nation of 

ours has fallen into !" Truly he is a person who tempts to 
digression. 

Dodson, in the preface of the Canon, acknowledges much 
assistance in the drawing up of the explanation from Robertson 
himself, not from Jones. The part which is worth dwelling on 
is what relates to Jones. If Dodson wrote a fair common English, 
the whole falls to the ground. His printed writings show nothing 
either odd or confused : but he may have got somebody to write 
them all. He could hardly have kept a composer for his own 
private letters; and I subjoin one to Robertson, which came into 
the hands of Dr. Hutton, from whom it passed to Dr. Olinthus 
Gregory, at whose sale I bought it. The reader is to see whether 
the meaning comes at once or whether he must read a sentence 
twice before he understands it. 

Sir. Being the other day taming over Mr. Simpson's new book, I 
took it in my head to try how much better his new approximations to the 
roots of equations were than those we commonly use, and find that his 
examples are packed, being such as our common operations will give to six 
or seven places the first substitution; which, with all his apparatus, he 
seldom exceeds above a figure or two. I determined therefore to reject his 
pretendect improvement and stick to the old way in the work I am putting 
together for Mr. Knapton [what this was I do not know] and set about 
composing that part of it. 

I believe you have found as well as I that these approximations are 
difficult to be worded so as that a person who cannot read algebra should 
readily understand and retain them [Dodson was very fond of expressing 
algebra in words, and did it with unusual precision and clearness] ; but it 
has happened that in this revision of the subject I have, by a little cooking 
of the old equation, happened upon the following approximation to the root 
of any surd, which I give you iu words that you may see how easy it will 
be to remember it. 

The number whose root is required I call the surd power. And the 
nearest similar real power, whether greater or less, I call the rational 
power. 

Multiply the rational power by the index more, one, and to the result 
add the product of the surd power by the index less one; reserving 
the sum. 



352 Some Account of James Dodson, F.R.S. [Oct. 

Also multiply the rational power by the index less one, and to the 
result add the product of the surd power by the index more one; reserving 
also this sum. 

Then as the first mentioned sum is to the second, so is the root of the 
rational power to the root of the surd power. 

I have sent you this in hopes it may come in time enough for the cube 
root in the arithmetical part of your navigation. And for that root it runs 
easier, thus. 

To twice the rational cube add the surd cube, and to twice the surd 
cube, add the rational cube. Then as the first sum is to the second, so is 
the root of the rational cube to the root of the surd cube. (Please 
turn over.) 

The investigation, being rather too long for a letter, I reserve till I see 
you unless yon desire it further, when I will transcribe and sent it. 

We have had a fortnight of very indifferent weather, but make shift to 
keep jogging on, and I am in great hopes the field-work may be finished 
before I am obliged to come to town : my next shall enclose the draft, 
which should have come now, but Sir Tho. is from home. I am, Sir, 
your obliged humble servant, J. Dodson. 
Sept. 17, 1752, by Act of Pari 1 style. 
[This was the fourth day of the new style.] 

And now for a letter from William Jones, which I happen to 
possess ; the man of influence and official station, who used small 
mathematicians like dogs ; and who was the corrector of Dodson's 
style. So far as one letter can go, it clears him of both 
imputations. It is to Hodgson, Dodson's predecessor but one, and 
is addressed on the outside " To Mr. James Hodgson, at Christ's 
Hospital, London, these presents." Hodgson's life was a counter- 
part of Dodson's : he was a private teacher and writer who ended 
in the mastership of the Royal Mathematical School. 

Honoured and beloved Sir. Tho Wednesday I came away I delivered 
the papers to your servant. It's my design to send them up in a little 
time, the calculations of problem (4) at large, so that everything may be 
evident to you as you proceed, without any trouble. I have altered the 
method from case(l) of Astronomic Problem (6) to case the (2) and hope 
to render it of more easy, universal and exact use. I will send one for the 
papers, and fairly insert problem (4) in writing among the others, and 
send them np to you without fail as soon as possible. I remain, most 
worthy Sir, your most obliged humble servant, W. Jones. 
Wantage, June 17, 1731. 

It is somewhat remarkable that so decided an instance of 
confused style should turn up, to set against the clearness of 
Dodson's writing. The reader asks how Jones could send up 
from Wantage the papers which he had left with Robertson's 
servant some Wednesday before : and he finds at last that " them " 
refers to papers spoken of afterwards. 

Dodson's criticism upon Simpson's method refers only to its 



1868.] Some Account of James Dodson, F.R.S. 353 

utility as a means of approximation, and is just : but neither 
Dodson nor Simpson himself saw its beauty as a theorem. As it 
is never mentioned in modern works I give it, without demon- 
stration, in modern symbols. 

To approximate to the small root of an equation, proceed as 
follows. Let the equation be c + CiX + c 2 x 2 + .... =0, and 
determine* D„ D 2 , &c., N 1} N 2 , &c, from 

c D 1 +c 1 =0, CoDi + cfii + Cz—O, c D 3 + c l D 2 + c 3 D l + c 3 =0, 

c N 1 + c 2 =0, 4^ + ^1 + 03=0, c N 3 +c 1 N 2 + c 2 N 1 + (? 4 =0, 

and so on. Then 

_ c qD»> 
CiD n — c N„ 

is the nearer to the root of the equation, the greater n is taken. 

Dodson's share in the projection of the Equitable is first 
mentioned in general publication by Nichols [Anecdotes, vol. v., 
p. 400). But the following extracts, with which I was favoured 
by Mr. Arthur Morgan, contain the whole account. 

In 1769 was circulated by the Directors a pamphlet entitled 
" A state of the Society for Equitable Assurances on Lives and 
Survivorships, and a state of facts from the year 1756 to the 
present time. Laid before the General Court the 28th of July, 
1769, by the Court of Directors." 

The following is an extract. 

1756. In this year Mr. James Dodson, having been refused admission 
to the Amicable Society on account of his age, determined to form a new 
Society upon a plan of assurance on more equitable terms than those of the 
Amicable, which takes the same premium for all ages. Having 
communicated this plan to several persons, they proposed to join him 
therein, if the intended Society could be established by Charter. The 
number of persons which engaged in this design were at first 55, and 
before they proceeded towards obtaining a charter, they set about providing 
a fund, and previous even to this consideration they held consultations 
about the plan of reimbursement and recompencc that should be made to 
Mr. Dodson and themselves. Accordingly it was determined that 15s. 
should be paid by every person making assurance with the said Society; 
5s. whereof should be paid to the said James Dodson for his life for his 
plans and trouble in planning the said Society, and making the necessary 
calculations; and the other 10s. were to go among the other persons [Raw 
beginners! primitive Christians! In our day this would be called omission, 
not commission: I never blushed for an ancestor until now.] The 

* I use D and N because they enter in the demonstration as denominator and nume- 
rator. I suppose the 1) and N of our commutation tables were chosen by Griffith Davies 

N 
from the part they play in — the first of the results wanted, and the suggesting formula. 

But this never struck me until now; and perhaps never struck some of my readers. 



354 Some Account of James Dodson, F.R.S. [Oct. 

application for a charter was conducted by Mr. Mores, and after three 
hearings before the Attorney and Solicitor General, to whom the petitioners 
were referred by his Majesty, a report was given against the petitioners. 
The petition having been presented at the Secretary of State's Office on the 
16th or 17th of April, 1757, and referred to the Attorney and Solicitor 
General, who did not make their final report until the 28th of July, 1761. 
In the mean time, that is to say on the 23rd of November, 1757, 
Mr. Dodson died. The hopes of a Charter being at an end, the generality 
of the original subscribers dropt the scheme, in the prosecution of which 
£600 had already been expended. [In the deed of settlement provision is 
made for the repayment of this money]. Mr. Edw. Howe Mores, however, 
and 16 more of the 55, resolved to persevere in establishing such a Society 
by deed, if it could not be done by Charter; and the present deed of 
settlement, of the 7th of September, 1762, was executed by every one of 
these 16 original Charter-fund proprietors. No table of calculations was 
procured till the 24th of January, 1764, and the Directors relying upon 
Mr. Mores for fixing every premium in the intermediate time. But at 
length such a table of lives was procured from the Executors of Mr. Dodson, 
and a resolution was put on the minutes for giving £300 to the children of 
Mr. Dodson as a recompence for the same. 

In a statement published and signed by Rich. Glyn, J. Sylvester, 
Win. Sclater, Edw. R. Mores, and Josiah Wallis, in reference to 
the Charter-fund, is found the following. 

The subscribers admit that in the year 1756, Mr. Dodson, not being 
able to obtain admission into the Amicable Society on account of his age, 
conceived a design of forming a Society upon the principle laid down by 
the late Dr. Halley, in his observations on the Breslau bills of mortality, 
viz. that the price of insurance on lives ought to be regulated by the age of 
the person upon whose life the insurance should be made. And that he, 
Dodson, caused to be inserted in the public papers an advertisement 
bearing date the 28th of February, 1756, giving notice of a meeting 
intended to be holden on the 2nd of March then next following, and 
desiring at that meeting the company of such gentlemen as niight be 
disposed to engage in such an undertaking. That they did accordingly 
meet upon the day appointed, and continued to meet weekly till the number 
amounted to about one hundred. 

Mr. William Morgan, in his ' account of the Rise and Progress 
of the Equitable Society/ gives the account of the finish of Dodson's 
connexion with the Society, as follows. 

Mr. Mosdell, who was stated to have been only an accountant, was 
appointed by the deed of settlement to be the first actuary, and on his death 
in December, 1764, [probably after six months trial, for the Equitable 
books show that the appointment is dated July 5, 1765,] Mr. [James] 
Dodson succeeded, who was the son of the excellent mathematician who 
computed the Society's tables, but without the mathematical learning of his 
father [ho was then just twenty-one years old, and the appointment must 
have been an acknowledgment of the father's services]. Upon obtaining a 
place in the Custom House more suitable to his abilities, Mr. Dodson 
resigned in April, 1767, when Mr. John Edwards was chosen .... 



1868.] Some Account of James Dodson, F.R.S. 355 

I will now give a few words to each of Dodson's works. I find 
them all mentioned in Watt's Bibliotheca, that is, all which I have 
ever seen; and I never heard of any others. And the heading is 
one of those short accounts which often occur in Watt, which 
could not be mended in the same number of words. "Dodson, 
James, F.R.S., an ingenious and very industrious mathematician 
in London/' 

The Anti-logarithmic Canon. Being a Table of Numbers, Consisting 
of Eleven Places of Figures corresponding to all Logarithms under 
100000. Whereby the Logarithm for any Number, or the Number 
for any Logarithm, each under Twelve Places of Figures, are readily 
found. With Precepts and Examples, showing some of the Uses of 
Logarithms, in facilitating the most difficult Operations in common 
Arithmetic, Cases of Interest, Annuities, Mensuration, &c. To 
which is prefix'd, An Introduction, Containing a short Account of 
Logarithms, and of the most considerable Improvements made, since 
their Invention, in the Manner of constructing them. By James 
Dodson. London: Printed for James Dodson, at the Hand and 
Pen in Warwick- Lane ; and John Wilcox, at Virgil's Head, 
opposite the New Church in the Strand. 1742. 
I should like to have a list of the authors who have shown their 
sense in the first words of the title of their first works : Dodson 
would find a place. The words " desiderandus videtur Canon 
Anti-Log arithmicus" were used by Wallis as far back as 1693. 
Young men very often think it is original-like, you know, to find 
their own phrases where good ones have been found by their fore- 
goers. There is an appendix of five pages, not mentioned in the 
title. " Of Decimal Notation, and its Use in solving Questions, 
which consist of Fractional Numbers by Logarithms." The work 
is dedicated to Lord Stanhope (Chesterfield). It was reviewed in 
the Works of the Learned for September, 1742, in so terse and 
accurate a way, and so free from eulogium, that I have no doubt 
the author wrote the article. 

There is a tangled story about the antilogarithmic Canon 
finished in manuscript by Warner and Pell. The utility of com- 
mon slanderers lies not in what they produce, but in what they 
omit : as to all of which there is a strong presumption that no 
means of constructing a story existed. If there had been a 
rumour, even a surmise, afloat that Dodson had seen this manu- 
script, Burrow would have got hold of it, and would have left it 
that Dodson had cribbed the work out of William Jones's library, 
and had published it as his own. And nothing but a very cautious 
comparison will show that he had not the opportunity so to do. 
For Collins's papers, in the bulk, came into the hands of William 



356 Some Account of James Bodson, F.R.S. [Oct. 

Jones, and were freely open to the crowd who had the liberty of 
his study : and Collins was certainly at one time the custodier 
of Warner's manuscript. No doubt a Canon with eleven hundred 
thousand computed figures, " elegant, in a large folio," would have 
been well known among the many mathematicians who haunted 
Jones's house and who knew what Wallis had written about it ; 
and its surreptitious publication would have required the, com- 
plicity of Jones and Robertson, at least, and the character of the 
transaction would have been known to many. But this is not all : 
it appears that Warner's manuscript, deposited with Collins to be 
restored on demand, actually did pass out of Collins's hands into 
those of Dr. Busby. It has never since been mentioned as seen. 
The authorities for the following collection of facts, Wallis, Pell, 
Thorndyke, and Collins are to be found in the Latin Algebra of 
Wallis {Opera, vol. ii., Alg. cap. xii.) ; the Macclesfield Correspon- 
dence (vol. ii., p. 197, 215, 219) ; and Halliwell's Letters on 
Scientific Subjects (Hist. Soc. Sci., pp. 80, 94, 95). 

Dr. Pell informed Wallis that Warner, assisted by himself, had 
finished a canon not long after 1631 : " about fifty years ago," 
says Wallis, which makes his writing to be near 1680, and very 
likely later. Wallis saw this canon, about 30 years before writing, 
say near 1650. In 1644, Pell, writing to Sir Chas. Cavendish, is in 
trouble about Warner's papers, the custodiers of which had become 
bankrupt, and he feared the papers had been or would be destroyed. 
We can only hope that poor Pell met all his troubles with as good 
heart as this one. 

In the mean time I am not a little afraid that all Mr. Warner's papers, 
anil no small share of my labours therein, are seazed upon, and most 
unmathematically divided between the sequestrators and creditors, who 
(being not able to ballance the account where there appeare so many 
numbers, and much troubled at the sight of so many crosses and circles in 
the superstitious Algebra and that blacke arte of Geometry) will, no 
doubt, determine once in their lives to become figure casters, and so vote 
them all to be thro wen into the fire, if some good body does not reprieve 
them for pye-bottoms, for which purposes you know analogicall numbers 
are incomparably apt, if they be accurately calculated. 

The papers were found, and in 1652, we find them in the pos- 
session of Dr. Thorndyke, prebendary of Westminster, who as the 
trustee and holder of Warner's papers, among which the full canon 
and an abridgment are particularly specified, writes to Pell to urge 
publication of the whole, and seems to admit that Pell has the 
copyright: a note by Pell, endorsed on the letter, states that 
publication is abandoned on account of incompleteness, not of the 



1868.] Some Account of James Dodson, F.R.S. 357 

Canon, but of " the papers." No more appears until December, 
1667, when we find the receipt given by Collins to Thorndyke, 
acknowledging the receipt of the Canon and other papers, to be 
restored on demand. In Sept., 1675, Collins writes to Tschirnhaus 
as follows : — 

Between the years 1630 and 1640 Dr. Pell and one Mr. Warner, 
deceased, mentioned in Mersennus, agreed to make a table of antilogarithms, 
which were to be called Antilogaritbmi Pellio-Warneriani; and accor- 
dingly such a table was computed, and left in the hands of Dr. Thorndyke, 
deceased, and cost Mr. Warner above 400 crowns the doing: as to the 
table itself it is a table of 99998 mean proportionals between an unit and 
100,000, each to eleven places of figures, elegant, in a large folio. . . . 

Thorndyke was dead, and Collins docs not say he had the table 
in his possession when he wrote : probably Thorndyke's executors 
found Collins's receipt and reclaimed the papers. Again, Collins, 
four years before, writing to James Gregory, March, 1671, gives 
the same account, as follows : — 

One Mr. Warner, deceased, whose Optics you find mentioned in Mer- 
sennus, did, about 32 years since, spend above an hundred pounds for aid, 
and took great pains himself, with some assistance from Dr. Pell, to calcu- 
late a table to twelve places of figures of 100,000 continual proportionals, 
to wit, to find 99999 mean proportionals between an unit and 100,000. 
Such a large table, elegantly writ, remains in the hands of Dr. Thorndyke, 
a prebendary of Westminster; the construction and uses of it, with the 
tactions of circles rendered analytical, were lent to one Gibson, deceased, 
in anno 1650, author of a book entitled Syntaxis Math., after whose death 
all his papers were consumed to light tobacco." (Maccl. Corr. ii. 219.) 

And again (p. 197, in a letter of which the date must be 
altered) " the tables I mentioned in Dr. Thorndyke's hands." So 
that the manuscript had gone back from Collins in 1671. It is 
passing strange that Collins, who was very well informed, and 
whose immense correspondence got him the name of the Attorney- 
General of the mathematics, should have been quite ignorant of 
Warner, Harriot's executor and the publisher of his very celebrated 
algebra, except as a person mentioned by Mersenne who, on like 
grounds, should have been the Procureur-Ge'neral. 

Wallis, when he wrote his note, not far from 1680, to which 
he put a last paragraph after 1685, says that Pell — who must 
have known all about it — told him the papers w^ere in the hands of 
Dr. Busby, of Westminster school, a very likely man to be the 
executor of the prebendary, and a very unlikely man to come by 
mathematical papers in any other way. When Pell made this 
communication to Wallis, he was meditating immediate publication, 
and his business was to ask Wallis to see the printing finished, in 
vol. xiv. 2 c 



358 Some Account of James Dodson, F.B.S. [Oct. 

case of his own death daring the proceeding; to this Wallis 
assented. In 1755, Dr. Birch procured for the Royal Society 
some of Pell and Warner's papers from the trustees of Dr. Busby. 
The antilogarithmic canon — I mean the manuscript, to avoid all 
mistake ; Dr. Busby himself was a canon, and probably an anti- 
logarithmic one — might, or might not, have been among them. 
Rigaud inadvertently writes that Birch procured "four large boxes" 
of these papers for the Society : but Birch only says that the Pell 
and Warner papers were mixed with Busby's papers in four large 
boxes : if these boxes exist, the canon may be in them still. But 
it strikes me as most likely that Pell, a man of energy and impulse, 
after arranging with Wallis, obtained possession of the manuscript 
with intent to publish immediately, and that it was mislaid at his 
death. He was a " shiftless man," and shirtless too, sometimes : 
he often wanted pen and paper; he was in the King's Bench not 
long after his conversation with Wallis ; and he died in poverty, 
and was buried at the cost of Dr. Busby. 

It is clear that Dodson had no opportunity of seeing Warner's 
manuscript in the possession of William Jones or any one else that 
we know of. But it would be strange if there were none to suspect 
that he got at it amosgepotically (that is, somehow or other) and 
made fraudulent use of it. A priori wisdom will find difficulties 
in any other hypothesis. Why should Dodson, of all persons, 
meditate so large an undertaking, and why an antilogarithmic 
canon rather than anything else? He knew Wallis's account, 
which he quotes ; and he might have seen Collins's letters in Jones's 
collection. What more easy than to suppose that he made a hunt 
for the manuscript ? Suppose him to have once been a West- 
minster boy — he must have gone to school somewhere — and to 
have made use of his knowledge to gain access to Busby's boxes ; 
what more is wanting? But though amusing myself with the 
love of evil which cannot help inventing all that is wanting to 
prove it, I am quite aware that it is open to anyone who can to 
trace the manuscript, and to examine the circumstances, in order 
to see whether — all apparent impossibility notwithstanding — 
Dodson found it and used it. It is quite certain that the fact of 
such a manuscript having existed must have been known in William 
Jones's circle : Wallis in print and Collins in the letters in Jones's 
library must have been referred to when Dodson published his 
Canon : and the acquisition of Busby's papers, in 1755, is pre- 
sumption that the possibility of obtaining the manuscript was 
recognized ; and not quite despaired of. 



1868.] Some Account of James Dodson, F.R.S. 359 

The only thing to be explained is the accordance of Warner 
and Dodson in extent of plan : five figures of argument and eleven 
of tabular result. The explanation is that both Warner and 
Dodson naturally aimed at making their tables coextensive with 
the great tables of Briggs and Vlacq. These had eleven figures ; 
we now say ten : but the characteristic formerly counted as a place 
of the logarithm. Both Warner and Dodson judged correctly that 
their table would lose much of its working value for high purposes 
if, going above seven figures, it were anything less than the nume- 
rical counterpart of the great tables, which must be used with it. 

Dodson very fairly quoted all he knew about Warner ; that is, 
he gave the passage from Wallis's Algebra of 1693, in English. 
But, apparently dissatisfied with the translation which had appeared 
in two editions of Sherwin's logarithms, he translated anew, refer- 
ring to Sherwin. 

I will here mention that the correction of misprints found in 
the copies of the Canon are in most cases in Dodson's own hand- 
writing. He followed this practice in more works than one. 

1747. Octavo (half sheets). The Calculator: being, correct and 
necessary Tables for Computation. Adapted to Science, Business, 
and Pleasure. By James Dodson, Accomptant, and Teacher of the 
Mathematics. London: Printed for John Wilcox, at Virgil! s Head, 
opposite the Neio Church in the Strand ; and James Dodson, next 
Door to the Blue Legg, near Bell-Dock, Wapping. m.dcc.xlvii. 

This work is dedicated to William Jones. Some copies have 
another title page, also of 1747, in which Wilcox alone is men- 
tioned in the imprint. This means that Wilcox took the risk off 
Dodson's hands within the year ; and thenceforward we no more 
find him publishing on his own account. 

With the exception of heavy calculators, to whom the Canon is 
occasionally useful — Benjamin Gompertz, for instance, who told 
me forty years ago he was always wanting it — this table is worth 
three of the Canon to anybody. Whoever can catch a copy should 
keep it. The table of binomial coefficients, up to the 34th power, 
is very useful. So is the table of specific gravities. The medley of 
coins, measures, regular solids and polygons, roots, logarithms, 
common, hyperbolic, logistic, trigonometrical, &c, interest, annui- 
ties, &c. &c, though not extensive, are great friends at a pinch. 
For a single book to travel with, and a good chance for anything 
that can be wanted, I know only Mr. Willich's table which can 
compare with it. But Dodson's two or three words to each head 
in the preliminary index enable the user to find his table in a 

2 c 2 



360 Some Account of James Dodson, F.R.S. [Oct. 

moment. The only additional friend mentioned is John Ellicot, 
F.R.S., the (in that day) celebrated watchmaker. 

1747-48, 1753, 1755, 12mo. The three volumes of the 
Mathematical Repository, by James Dodson, Accountant and 
Teacher of the Mathematics. The full titles will recal the contents : 
they describe the volumes as 

(i.) Containing analytical Solutions of near five hundred questions, 
mostly selected from scarce and valuable authors. Designed as examples 
to Maclaurin's and other elementary books of algebra; and to conduct 
beginners to the more difficult properties of numbers. 

(ii.) Containing algebraical solutions of a great number of problems, 
in several branches of the mathematics. I. Indetermined questions, 
solved generally, by an elegant method communicated by Mr. De Moivre. 
II. Many curious questions relating to chances and lotteries. III. A great 
number of questions concerning annuities for lives, and their reversions; 
wherein that doctrine is illustrated in a multitude of interesting cases, with 
numeral examples, and rules in words at length, for those who are un- 
acquainted with the elements of these sciences, &c. 

(iii.) Containing analytical solutions of a great number of the most 
difficult problems, relating to annuities, reversions, survivorships, insurances, 
and leases dependent on lives; in which it has been endeavoured to exhaust 
the subject. 

All is 'printed for John Nourse, at the Lamb, opposite 
Katherine Street in the Strand/ The dedications are to De Moivre, 
David Papillon, F.R.S., and Lord Macclesfield and the Council of 
the lloyal Society. There was a second edition of the first volume 
in 1775 ; I am not aware of any other editions of the remaining 
volumes. *I should think there were none, for the remaining stock 
of the work was locked up by some of the incidents of trade, and 
was let out about 35 years ago, when the market was suddenly 
supplied with uncut copies. 

1750, 4to. The Accountant, or the method of book-keeping, deduced 
from clear principles, and illustrated by a variety of examples. By 
James Dodson, Teacher of the Mathematics. London printed for 
J. Nourse at the Lamb opposite Kathcrine-Street in the Strand. 

This book is dedicated to Lord Macclesfield, whose accounts 
Dodson seems to have been employed in, and who, it is hinted, 
desired that double entry should be applied to the business of an 
estate and of a farm. The work also applies it to retail trade, a 
thing till then unexemplified : and the shoemaker's trade is chosen 
on account of the variety of his transactions. This book is 
excessively scarce : the copy in the Museum and my own being 
the only ones I ever heard of. 

1751. 8vo. In this year Dodson published an enlarged edition 
of Wingate's Arithmetic. The preface is dated April 4, 1751. 



1868.] Some Account of James Dodson, F.R.S. 361 

It was reprinted several times. I have only seen the edition of 
1760. Wingate is the best of the old writers, greatly superior to 
Cocker (or rather Hawkins) : and Dodson's are the best editions 
of Wingate, according to Watt — and myself. Wingate and Cocker 
were the two household-gods of arithmetic. In 1750, Arthur 
Murphy introduced their names upon the stage in ' the Apprentice/ 
Wingate as an old merchant who is constantly recommending 
Cocker : and I believe that this is the way in which Cocker became 
a bye-word ; I can find nothing earlier. 

4to. (pp. 18). An account of the Methods used to describe lines, 
on Dr. Halley's Chart of the Terraqeous Globe; showing the 
Variation of the Magnetic Needle about the Year 1756, in all the 
known Seas; their Application and Use in correcting the Longitude 
at Sea; with some Occasional Observations relating thereto. By 
William Mountaine and James Dodson, Fellows of the Royal Society. 
London: Printed for W. and J. Mount, T. Page and Son, on 
Tower Hill. 1758. 

Of Mountaine I only know that he was one of the founders of 
the Equitable, and that he was Dodson's executor. Watt calls 
this tract a folio, and gives it a first edition in 1718. The truth, 
as appears by the tract itself, is that in 1744 the two collected 
observations from the Admiralty, the India and African companies, 
and private communications. On these data they published a 
chart in 1745, which I have never seen. By this chart Dodson 
must have been known as having paid attention to matters con- 
nected with navigation, a circumstance which may have facilitated 
his appointment to the B. M. School. 

To the preceding list must be added three papers in the 
Philosophical Transactions; 1752, p. 333, on the improvement of 
the bills of mortality; 1754, p. 487, on annuities and survivor- 
ships ; 1753, p. 273, on logarithmic series. The second and third 
papers are written to show how to dispense with the use of fluxions, 
which all the mathematicians who could were very apt to intrude 
into every part of algebra above the merest elements. This prac- 
tice did much harm : the packing up of all the difficulties of series 
into the abbreviations of the differential calculus was a fearful 
drawback on the rigour of the science. It is only in our own day 
that mathematicians have become alive to the danger of all sorts 
and conditions of interminable series. Here is an instance for the 
reader of the mathematical part of this Journal. Take the series 

4-3# 16-15# 3 36-35a , 
1.2 iV+ 3.4 * + 5.6 *+•••• 



362 Some Account of James Dodson, F.R.S. [Oct. 

This is certainly convergent when x is < or =1. When x=l, 
it seems to be _+_+_+.... or l- -+---+- 

— a +...., or hyp. log 2. But it is not : what is it, then ? 

The algebra of annuities, &c., was put into working form by 
De Moivre, Simpson, and Dodson, who gained the necessary 
restraint upon themselves by having been occupied in the actual 
practice of the subject. It is almost a rule that a writer on any 
mixed mathematical subject who has not been actually engaged in 
mixture overdoes the mathematical part : I do not mean that he 
introduces mathematics where it ought not to be — this he may or 
may not do — but that he makes too much of mathematics where 
some ought to be. De Waring, one of the most useful algebraical 
discoverers of the century, made a great failure in an attempt to 
write on the subject : and as the history of his tract is peculiarly 
matter for this Journal, I will end with it. 

The book was called ' On the principles of translating algebraic 
quantities into probable relations and annuities, &c. By E. Waring,' 
Cambridge, 1792, 8vo. (pp. 59). It would have sold well if the 
implied title-promise had been kept : it is not every one who can 
translate algebraic quantities into an annuity, or into a probable 
relation with the chance of a reversionary legacy. As it was, no 
book ever fell more dead from the press : it is not mentioned by 
any of Waring's biographers before 1815. Some notice of it was 
taken in the first edition of Hutton's Mathematical Dictionary, 
vol. ii. p. 276, which induced Mr. Baily to write to Hutton for 
information. Hutton answered that he had never seen nor heard 
of the tract ; that the account in which it appeared was furnished 
by Waring himself, whom he took to be good authority for a work 
of his own : that it certainly was not one of the pamphlets which 
passed between Waring and Powell during the contest for the 
professorship; but that he had found a "referment" to it in 
Wood's Algebra. Baily accordingly wrote to Dr. Wood, who in 
answer gave the title, and offered to lend his own copy (Sept. 5, 
1808). The offer was accepted; for on the 15th Baily wrote to 
Hutton a short account of the work, which he described in stronger 
terms than he afterwards used in his book on assurances 
(Pref. p. xx.), and in which I quite agree with him. " Certainly 
he has thrown no new light on the subject. His problems (if a 
string of detached observations are worthy of that name) are quite 



1868.] Some Account of James Dodson, F.R.S. 363 

elementary, and his loose and illogical [he meant immethodicaW] 
method of treating them adds neither grace nor dignity to the 
subject. The very title of the book betrays the inaccuracy of his 
style." Baily afterwards picked up a copy for himself. When I 
came to look after this book, about 1835, 1 could find no mention 
of it : and I asked Mr. Baily to lend it to me. He could not find 
it ; and I ventured to express a suspicion that he had mistaken the 
author's name. Whereupon he producedf what he knew where 
to find at once, the bookseller's receipt, which stated name and 
title. It turned up when his books were arranged for sale, and I 
bought it. Some time afterwards I found that the library of 
Queen's College, Cambridge (which was not Waring's College), 
contained some half dozen copies. A few of these were, upon 
representation of the state of the case, presented to other libraries, 
I forget which: probably the Royal Society or the British Museum 
will now possess the book. 

I have never had so strongly impressed upon me the littleness 
of the period preceding the accession of Geo. III. We do not 
make much boast of its collective literature, and yet it was the day 
of Mansfield, Fielding, Sam. Johnson, David Hume, Sterne, Gray, 
Garrick, Blair, Hor. Walpole, Smollett, Robertson, Adam Smith, 
Blackstone, Joshua Reynolds. In applied science there was no 
great strength : but in pure mathematics we have little more than 
the remnants of a stronger period : some good names, but far too 
few to count as a school, belong especially to the time. Its 
historical masterpiece is the Biogr. Philosoph. of Benjamin Martin 

* In reply to a suggestion whether unmethodical would not he the preferable word, 
Mr. De Morgan writes : — " I made the word unmethodical, upon the old analogy. Un is 
Saxon ; and properly belongs to Saxon words, as unaware, unbeaten. In and Im are for 
Latin; though certainly the Saxon has intruded, as in ungovernable, unsophisticated, 
uncommon, &c. But the great bulk of our Latin words still keep im or in, according to 
the consonant which follows, as imperceptible, immense, innocent, and a crowd of others. 
On looking for immethodical in a little sixpenny Johnson of the stalls, — there it is. I 
generally consider the foreign dictionaries as good authorities as to English words : and 
in the French, German, and Italian which I keep at hand, I find the word in all. I 
find capricious cases; as interminable and unterminated, indeterminate and undetermined 
(of which the mathematicians have availed themselves). Also insatiate and unsatiated. 
The rule seems to be that when the Saxon ed is at the end, the Saxon «» shall be at the 
beginning; aud Latin, Latin. This may be called the sandwich rule, if it be a rule. 
The end of it is that any one may do as he pleases, which is the glory of English." — 
Ed. J. J. A. 

f Francis Baily was a paragon of method : he practised and enforced. I found him 
one day in the act of finishing a note, which he showed me; it was before the time of 
prepaid letters. One of those tradesmen who, when a customer is as good as the bank, 
persist in making a banker of him during convenience, would not send in his bill. The 
note ran as follows : — " (No. 1 ). Sir, — I beg you will oblige me by sending in your 
account forthwith. Yours, F. Baily. P.S. This notice will be repeated once a week 
until it is complied with." No. 2 was not wanted : the tradesman declined to grant 
His Majesty an annuity of 8s. 8</., payable weekly. 



364 On the Application of Bonuses [Oct. 

(1761), a work of unmatched inutility. And yet good biography 
had commenced in force with the Biogr. Brit, in 1747. The 
total absence of historical effort encouraged the learned vicar of 
Twickenham, George Costard, to give to his work on the globes, 
(1767), full of every kind of miscellaneous historical statement, the 
title ' History of Astronomy.' All my reading has led me to 
suspect that the doubts and dangers of the disputed right to the 
Crown, which lasted from the rising of 1715 to that of 1745, 
produced a paralysing effect upon the intellectual energies of the 
country. 



On the Application of Bonuses to limit either the term of an 
Assurance or the number of payments to be made under it : being 
Excerpts from a paper read before the Actuarial Society of Edin- 
burgh, on 5th December, 1867. By James R. Macfadyen, 
of the City of Glasgoxv Life Assurance Company. 

1 HE application of bonus to the hastening of the time fixed by 
the original contract for the payment of a life assurance policy, 
and the using of it to extinguish the premiums payable during the 
latter years of life, are systems, which, partly from their growing 
favor with the public, and consequent adoption by many offices, 
and still more from the fact, that the subject is by no means 
exhausted, notwithstanding various able articles that have appeared 
in this Journal, seem to me to render apology unnecessary for 
again treating of it. 

The papers to which I allude as having already taken up the 
matter, are two articles by Mr. Sprague in the sixth volume, 
embracing both branches of the subject, and a more recent letter 
by Mr. Marr, (vol. xiii. p. 246) referring to the first of these two 
systems alone. 

Though, as we shall hereafter see, these systems of applying 
bonus are very closely allied in character, and they have thus been 
coupled in this paper, it will be more convenient in the first place 
to confine ourselves to that in which the bonus is applied to limit 
the term of insurance only. 

The problem usually arising in this case is to find the term of 
the Endowment Assurance into which a given cash bonus will 
convert an ordinary assurance. Mr. Sprague solves this problem 
by making an equation at the age the change is effected between 
l t>, the cash bonus then allocated to the policyholder, which is the